Score : $1000$ points

Problem Statement

Takahashi and Snuke came up with a game that uses a number sequence, as follows:

• Prepare a sequence of length $M$ consisting of integers between $0$ and $2^N-1$ (inclusive): $a = a_1, a_2, \ldots, a_M$.
• Snuke first does the operation below as many times as he likes:
  • Choose a positive integer $d$, and for each $i$ $(1 \leq i \leq M)$, in binary, set the $d$-th least significant bit of $a_i$ to $0$. (Here the least significant bit is considered the $1$-st least significant bit.)
• Choose a positive integer $d$, and for each $i$ $(1 \leq i \leq M)$, in binary, set the $d$-th least significant bit of $a_i$ to $0$. (Here the least significant bit is considered the $1$-st least significant bit.)
• After Snuke finishes doing operations, Takahashi tries to sort $a$ in ascending order by doing the operation below some number of times. Here $a$ is said to be in ascending order when $a_i \leq a_{i + 1}$ for all $i$ $(1 \leq i \leq M - 1)$.
  • Choose two adjacent elements of $a$: $a_i$ and $a_{i + 1}$. If, in binary, these numbers differ in exactly one bit, swap $a_i$ and $a_{i + 1}$.
• Choose two adjacent elements of $a$: $a_i$ and $a_{i + 1}$. If, in binary, these numbers differ in exactly one bit, swap $a_i$ and $a_{i + 1}$.

There are $2^{NM}$ different sequences of length $M$ consisting of integers between $0$ and $2^N-1$ that can be used in the game.

How many among them have the following property: if used in the game, there is always a way for Takahashi to sort the sequence in ascending order regardless of Snuke's operations? Find the count modulo $(10^9 + 7)$.

Constraints

• All values in input are integers.
• $1 \leq N \leq 5000$
• $2 \leq M \leq 5000$

Input

Input is given from Standard Input in the following format:

$N$ $M$

Output

Print the number, modulo $(10^9 + 7)$, of sequences with the property: if used in the game, there is always a way for Takahashi to sort the sequence in ascending order regardless of Snuke's operations.

2 5

352

Consider the case $a = 1, 3, 1, 3, 1$ for example.

• When the least significant bit of each element of $a$ is set to $0$, $a = 0, 2, 0, 2, 0$;
• When the second least significant bit of each element of $a$ is set to $0$, $a = 1, 1, 1, 1, 1$;
• When the least two significant bits of each element of $a$ are set to $0$, $a = 0, 0, 0, 0, 0$.

In all of the cases above and the case when Snuke does no operation to change $a$, we can sort the sequence by repeatedly swapping adjacent elements that differ in exactly one bit. Thus, this sequence has the property: if used in the game, there is always a way for Takahashi to sort the sequence in ascending order regardless of Snuke's operations.

2020 530

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